#119073

order	512
nilpotency class	3
isoclinism	isoclinic class of #119068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

479 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,11,7,10,5,21)(3,12,16,27,14,8,15,23)(4,9,20,26,18,25,19,28)(13,22,30,24,31,29,17,32)(33,34)(35,44)(36,49)(37,54)(38,42)(39,43)(40,46)(41,52)(45,60)(47,56)(48,59)(50,55)(51,58)(53,63)(57,62)(61,64);(1,3,7,14)(2,8,10,12)(4,17,18,30)(5,16,6,15)(9,24,25,32)(11,27,21,23)(13,20,31,19)(22,28,29,26)(33,35,39,48)(34,40,43,56)(36,50,52,61)(37,47,38,46)(41,57,49,60)(42,59,54,44)(45,51,62,53)(55,63,64,58);(1,4)(2,9,10,25)(3,13)(5,20)(6,19)(7,18)(8,22,12,29)(11,28,21,26)(14,31)(15,30)(16,17)(23,32,27,24)(33,36,37,51,39,52,38,53)(34,41,42,58,43,49,54,63)(35,45,46,61,48,62,47,50)(40,55,44,57,56,64,59,60)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26269);